Lesson 7 of 9
The average that lies
Caterpillar's revenue grew 7.41 percent a year on average and 6.44 percent compounded. Only one of those two numbers describes what happened.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- The arithmetic mean of yearly growth rates is always at least as high as the compound rate, and higher whenever the years differ.
- Only the compound rate reproduces the ending figure from the starting one.
- The gap grows with the spread of the yearly changes, so the more volatile the series, the more the average overstates.
Average nine annual growth rates and you get a number. Compound the same nine years and you get a different one. The first is arithmetically correct and describes nothing that happened. This is easiest to see on a real series.
Caterpillar revenue, fiscal 2016 to fiscal 2025
- Revenue
Show the numbers
| Revenue | |
|---|---|
| FY16 | 38,537 |
| FY17 | 45,462 |
| FY18 | 54,722 |
| FY19 | 53,800 |
| FY20 | 41,748 |
| FY21 | 50,971 |
| FY22 | 59,427 |
| FY23 | 67,060 |
| FY24 | 64,809 |
| FY25 | 67,589 |
The two averages
The arithmetic mean adds the yearly growth rates and divides by how many there are. The geometric mean, which is the compound annual rate, is the single rate that would have taken the first year's figure to the last one.
Caterpillar's revenue went from 38,537 million in fiscal 2016 to 67,589 million in fiscal 2025, nine growth years apart. That ratio is 1.7539, and the ninth root of it is 1.0644. The compound rate is 6.44 percent a year.
What the average says instead
Why the higher one is wrong
Take the starting revenue and grow it at 7.41 percent for nine years. That gives 73,344 million. The company filed 67,589. The average overstates the destination by 5,755 million, which is 8.5 percent of the real figure, on a series where nothing was estimated and every number came out of a 10-K.
The cause is that percentage changes do not add. A year down 22.4 percent and a year up 22.1 percent average to roughly zero and leave you below where you started, because the recovery applies to a smaller base. Any series with variation has this property, and a series without variation has the two averages equal.
How big the gap gets
The standard deviation of Caterpillar's nine growth rates is 14.60 percent, and the gap between the two averages is 0.97 points. That relationship is general: the gap rises roughly with the square of the spread, so doubling the volatility of a series roughly quadruples the amount by which its arithmetic mean overstates what an owner actually got.
The practical rule is short. When a figure describes what happened, use the compound rate. Any published growth rate should state which one it is, and a growth rate that cannot reproduce the ending figure from the starting one is the wrong average.
Check yourself
A holding falls 50 percent in year one and rises 50 percent in year two. What is the compound annual rate?
- Zero
- Negative, about minus 13.4 percent a year
- Positive, about 4 percent a year
Show the answer
Answer: B.
0.5 multiplied by 1.5 is 0.75, so 100 becomes 75 over two years. The square root of 0.75 is 0.866, which is a compound rate of minus 13.4 percent a year. The arithmetic mean of the two years is zero and it describes nothing.Which average always reproduces the ending value from the starting value?
- The arithmetic mean of the yearly rates
- The geometric mean, which is the compound annual rate
- Both, if the period is long enough
Show the answer
Answer: B.
The compound rate is defined as the one that does exactly that. The arithmetic mean only matches it when every year's change is identical.Two series have the same compound growth rate. One is far more volatile. What is true of their arithmetic means?
- They are the same
- The volatile one has the higher arithmetic mean
- The volatile one has the lower arithmetic mean
Show the answer
Answer: B.
The gap between the two averages widens with the spread of the yearly changes, so at equal compound growth the more volatile series reports the higher average.Sources
- Caterpillar Inc, annual reports on Form 10-K for fiscal years 2016 to 2025, us-gaap Revenues as filed. Fiscal 2016 from accession 0000018230-17-000041; fiscal 2025 from accession 0000018230-26-000008, filed 13 February 2026.
- Damodaran, Investment Valuation: the distinction between arithmetic and geometric average returns and when each is appropriate.
What this leaves out
- This is Caterpillar's own filed revenue, not a return to any investor. It is used because it is real, public and traceable to a document.
- Revenue in a fiscal year can be restated later. The figures here are as filed in each year's own 10-K, and fiscal 2016 through 2024 were also carried forward unchanged in the following year's filing.
- An arithmetic mean is the right estimate for a single future year drawn at random from the distribution. It is the wrong description of a realised multi-year result.
- Nothing here is advice about any security.